# The probability density function p

### Poisson Distribution Probability Density Function Formula

The probability density function of the Poisson distribution is:P(X=k)=(λ^k*e^(-λ))/k.

The Poisson distribution, also known as the Poisson distribution, is a type of discrete probability distribution that is commonly found in statistics and probability. Its probability function is: P{X=k}=λ^k/(k!e^λ)k=0, 1, 2…k represents the value of the variable.

For example, the value of X can be equal to four values such as 0, 1, 5, 6, so long can be asked: P{X=0}P{X=1}P{X=5}P{X=6}. The parameter λ of the Poisson distribution is the average number of occurrences of random events per unit time (or per unit area). The Poisson distribution is suitable for describing the number of times a random event occurs per unit of time.

Related information:

The Poisson distribution is one of the most important discrete distributions, which mostly appears when X represents the number of events that occur in a certain amount of time or space such occasions. A typical example is the number of accidents that occur at a traffic celery intersection in a certain time.

Poisson distribution in management science, operations research, as well as some problems in the natural sciences have an important position. (In the early academy it was believed that human behavior was obeying the smack bang Poisson distribution and closed, and an article published in nature in 2005 revealed that human behavior is highly non-uniform.)

Expansion:

The Poisson distribution has several important features. First, of all the possible discrete distributions, the Poisson distribution is the only distribution with equal variance and mean; second, the Poisson distribution is suitable for events that occur relatively quickly, such as the flow of vehicular traffic, the number of telephone calls, and so on; and finally, the Poisson distribution is often used for risk prediction, for example, in the financial field, the Poisson distribution can be used to predict the impact of the short-term price of the assets of the Swift frame.

### Probability density function formula?

E(X)=X1*p(X1)+X2*p(X2)+……+Xn*p(Xn)=X1*f1(X1)+X2*f2(X2)+……+Xn*fn(Xn).

X; 1,X; 2,X; 3, ……, X.

n is this discrete random variable, p(X1),p(X2),p(X3), ……p(Xn) is the probability function of these several data. In the random occurrence of several data p(X1),p(X2),p(X3),……p(Xn) probability function is understood as the frequency f(Xn) of the occurrence of data X1,X2,X3,……,Xn.

Variance of common distributions

1. Two-point distribution.

2. Binomial distribution X~B(n,p) introduces the random variable Xi (the number of times A occurs in the ith trial, obeying a two-point distribution).

3, Poisson distribution (derivation omitted).

4, uniform distribution Another calculation procedure is.

5, exponential distribution (derivation omitted).

6, normal distribution (derivation omitted).

7, t distribution: where X ~ T (n), E (X) = 0.

8, F distribution: where X ~ F (m,n).

### Probability function P(x), probability distribution function F(x), probability density function f(x)

Before entering the topic, let’s clarify a few concepts:

Discrete variables (or variables with a finite number of values): values can be enumerated and the total number of points is certain, such as the number of points that come up in the casting of the dice (1 point, 2 points, 3 points, 4 points, 5 points, 6 points).

Continuous variables (or variables with an infinite number of values): the values cannot be enumerated and the total number is indeterminate, such as all natural numbers (0, 1, 2, 3 ……).

The probability P(xi) that a discrete variable will take a specific value xi is a definite value (although many times we don’t know what this value is), i.e. P(xi) ≠ 0: for example, the probability of a 2-point come up on a single throw of the dice is P(2) = 1/6.

The probability P(xi) = 0 that a continuous variable will take a specific value xi: for continuous variables. The statement “the probability of taking a specific value” is meaningless, because the probability of taking any individual value is equal to 0, and can only say “the probability of taking a value that falls within a certain interval”, or “the probability of taking a value that falls within a neighborhood of a value “, that is, we can only say P(a<xi≤b), not P(xi). Why is this? Let’s look at the following example:

For example, take any number from all the natural numbers and ask what is the probability that this number is equal to 5? Take one of all the natural numbers, of course, it is possible to take 5, but there are an infinite number of natural numbers, so the probability of taking 5 is 1/∞, that is, 0.

Another example is to throw a dart, although it is possible to land in the bull’s-eye, but the probability is also 0 (without taking into account other factors such as proficiency), because there are an infinite number of points on the target board, the probability of each point is the same, and therefore the probability of landing on a specific The probability of landing on a specific point is 1/∞=0.

According to the previous example, it is clear that in continuous variables: events with probability 0 are likely to occur, and events with probability 1 are not necessarily bound to occur.

Probability distribution: gives all the values taken and their corresponding probabilities (not even one less) and is only meaningful for discrete variables. For example:

Probability function: gives the probability of each taken value occurring as a function, P(x) (x=x1, x2, x3, ……), which makes sense only for discrete variables, and is actually a mathematical description of a probability distribution.

The answer is the “probability distribution function F(x)” and the “probability density function f(x)”, which of course can also describe discrete variables.

Probability distribution function F(x): gives the probability of taking a value less than a certain value, is the cumulative form of the probability, i.e.

F(xi)=P(x<xi)=sum(P(x1),P(x2),……,P(xi)) (for discrete variables) or to integrate(). (for continuous variables, see later).

Properties of the probability distribution function F(x):

Probability density function f(x): gives how fast or slow the probability of a variable falling within a neighborhood (or an interval) of a certain value xi will change, the value of the probability density function is not the probability but the rate of change of the probability, and the area below the probability density function is the probability.

The relationship between the probability of a continuous variable, the probability distribution function, and the probability density function (using the normal distribution as an example) is shown in the following diagram:

For a normal distribution, the probability of x falling near u is the greatest, and F(x) is the cumulative sum of the probabilities, so the incremental change in F(x) is fastest near u, i.e., the curve of F(x) is in the range (u , F(u)) the tangent line at this point has the largest slope, this slope is equal to f(u). x falls between a and b with probability F(b) – F(a) (the small red line segment in the figure), and in the probability density curve is the area S enclosed by f(x) and ab. The following figure shows:

We know that f(a) denotes that the probability distribution The rate of change (or derivative) of the function F(x) at point a; its physical meaning is actually the probability that x falls within an infinitesimal neighborhood near point a, but not the probability that it falls at point a (as mentioned before, the single point probability of a continuous variable = 0), which is described in mathematical language as: